1-s2.0-0307904X9390072O-main

Analysis of fluid storage tanks including foundation-superstructure interaction

A. R. Kukreti, M. M. Zaman, and A. Issa

School of Civil Engineering and Environmental Science, University of Oklahoma, Norman, OK, USA

An analytical formulation is developed to predict the flexural behavior of a cylindrical liquid storage tank resting on an isotropic elastic soil medium, which is modelled as a half space. The interface between the plate foundation and the soil medium is considered to be smooth and continuous. The plate deflection function is assumed in the form of a power series expansion in terms of the radial coordinate. The procedure accounts for the interactions between the tank wall and the plate foundation, and between the plate foundation and the soil medium. The principle of minimum potential energy is used to evaluate the unknown coefficients appearing in the assumed power series expansion and also the unknown interacting forces at the tank wall-plate foundation junction. Any number of terms can be considered in the assumed dejection function. Analytical expressions are obtainedfor the plate foundation deflections and radial moment, the contact stress distribution, the tank wall displacements, and the tank wall stress resultants. The results obtained compare well with the finite element analysis of a similar problem. Results of a parametric study are also presented to demonstrate the efSect of the various geometric and material parameters on the flexural behavior of the system.

Keywords: tank, foundation, soil, interaction

1. Introduction

An analysis of the interaction between a structure and the underlying soil medium is of significant interest to both structural and geotechnical engineers. Most of the analytical and experimental studies reported in this area have considered the interaction between the foundation and the soil, but only a few have included the interaction between the foundation and the superstructure. Fluid storage tanks are an example of such a class of structures, in which the effect of foundation-superstructure interaction may be significant. In conventional analysis methods, the tank wall and the foundation-soil system are treated separately. Some convenient boundary conditions are assumed for the top and the bottom edges of the tank wall (usually fixed boundary conditions are assumed at the bottom), and it is analyzed as an independent component of the total system. The base reactions computed for the tank wall and the gravita- tional loads are then transferred to the foundation-soil system, which is then analyzed separately. Thus, such an idealization ignores the structural stiffness transmitted to the foundation from the tank wall in the form of bending

Address reprint requests to Prof. Kukreti at the School of Civil Engineering and Environmental Science, University of Oklahoma, 324 W. Boyd St., Room 327F, Norman, OK 73019, USA.

Received 6 February 1992; revised 27 April 1993; accepted 25 May 1993

618 Appl. Math. Modelling, 1993, Vol. 17, December

moment and shear force. rigidity of the foundation concentration of moment

. ^

Depending on the relative and the tank wall, a high may occur at the junction

between the foundation and the wall. To analyze and predict the behaviour of the plate

foundation-soil system alone, an extensive amount of literature is available. The notable methods suggested include the works of Borowicka,’ Zemochkin,’ Ishkova,3 Brown4 Selvadurai,’ Faruque and Zaman6 and Zaman, Kukreti, and Issa.7 Abundant literature is available for cylindrical storage tanks covering classical analysis*-” and design techniques. i1,i2 Booker and Small’j have presented a flexibility-type analysis procedure, which accounts for the interaction between the tank wall and the plate foundation-soil system by satisfying the com- patibility conditions at the plate foundation-tank wall junction. Cheung and Zienkiewicz,14 Smith,” and Mah- mood16 have applied the finite element method to analyze such soil-structure interaction problems. But, the use of the finite element method requires each structural configuration to be solved independently, which is computationally expensive.

In this paper an analytical formulation is presented that can be used to predict the flexural behavior of cylindrical fluid storage tanks, having circular plate foundations and resting on an isotropic elastic half space. The formulation accounts for the interaction between the tank wall and the plate foundation as well as the interaction between the plate foundation and the elastic half

0 1993 Butterworth-Heinemann

Analysis of fluid storage tanks: A. R. Kukreti et al.

space. Analytical expressions are developed for the following: contact stress distribution at the interface between the plate foundation and the elastic half space; plate foundation deflections and moments; and tank wall deflections and internal forces.

The analytical formulation presented in this paper is based on the principle of minimum potential energy. The deflection function describing the behavior of the plate foundation is approximated by a power series expressed in terms of the radial coordinate. The unknown coefficients appearing in this power series and the bending moment and shear force acting at the junction of the tank wall and plate foundation are treated as the generalized coordinates. The boundary conditions at the plate foundation-tank wall junction are used to reduce the number of independent generalized coordinates by expressing some of them in terms of the others. The potential energy functional of the total system is evaluated based on the following contributions: strain energy of the soil medium; strain energy of the circular plate foundation; strain energy of the tank wall; work done by the vertical fluid pressure acting on the plate foundation; and the work done by the horizontal fluid pressure due to tank wall deformations. The total potential energy functional, so obtained, is minimized with respect to each independent generalized coordinate. The resulting set of linear simultaneous equations is solved to obtain the values of the unknown generalized coordinates. Finally, these values are substituted in the assumed deflection function, and the analytical expressions for the previously mentioned quantities are obtained. Compared with other numerical techniques, such as the finite element method, the analysis procedure presented in this paper is computationally efficient and inexpensive. For an example problem, the results obtained are compared with those reported by Booker and Small.’ 3 Findings of a parametric study undertaken to investigate the effect of geometric and material-related parameters on the interaction between the tank wall and the plate foundation and between the plate foundation and the soil medium are also reported.

It may be noted that in the present analysis the soil medium is treated as an elastic material for simplicity and to keep the proposed analysis technique mathematically manageable. Nonlinearity of soil is likely to influence the response and the interaction between various components of the tank-foundation-soil system. The extent of such influences would depend on the degree of nonlinearity of the soil and the relative rigidity of the system. The soil moduli generally decrease with increasing load or stress level. The resulting vertical displacement of a foundation plate including nonlinearity of soil is, therefore, expected to be higher than that obtained by considering soil as an elastic material. Effect of soil nonlinearity on the magnitude and distribution of bending moment would be dictated by the change in the relative displacement of the plate. Because soil moduli generally increase with depth, the effects of soil nonlinearity are expected to be less significant because the degree of nonlinearity is expected to decrease with increasing depth.

SOk MEDl”l.4 @s, “$1

Figure 1. Schematic representation of the cross-section liquid storage tank system.

of a

2. Analytical formulation

A cross-section of the wall-plate foundation-soil system analyzed herein is shown in Figure 1. The radial coordinate r is measured from the plate foundation center. The displacement of the plate foundation-soil interface is denoted by w(r) and is measured in the z-direction, defined positive toward the soil medium. The height of the liquid in the tank is measured from the plate foundation surface. Any point in the liquid medium is located by the x-coordinate, measured from the top surface of the plate foundation, and the y-coordinate (horizontal), measured from the middle surface of the cylindrical tank wall (positive outward). No cover is considered at the top of the tank, and the upper edge of the wall is assumed to be free so that all forces at this edge vanish. The unit weight of the liquid in the tank is denoted by y and the liquid height in the tank is denoted by d, which is also considered the height of the tank wall. The plate foundation and tank radius are denoted by a. The plate foundation and tank wall thicknesses are denoted by t, and t,, respectively. The Young’s moduli and Poisson’s ratios of the soil, plate foundation, and tank wall materials are defined as (E,, vJ, (E,, v,), and (E,, v,), respectively. Throughout this paper the subscripts s, p, and t are used to refer to the soil, the plate foundation, and the tank wall, respectively. The free body diagrams of the tank wall, the plate foundation, and the soil medium are shown in Figure 2. The bending moment, M,, and shear force, Q,, at the junction between the tank wall and the plate foundation are considered unknowns and are treated as two additional generalized coordinates.

The formulation presented in this paper is limited to an axisymmetrical case. The tank wall and plate foundation materials are assumed to be linear elastic. The thin shell and the thin plate bending theories are used to describe the flexural behavior of the tank wall and the plate foundation, respectively. The soil medium is modelled as an isotropic elastic half space region. The interface between the plate foundation and the half space region is assumed to be smooth and continuous. This implies that only the normal component of the contact stress is present at the interface, and the interface displacements are uniquely described by either the plate foundation deflections or the soil surface deformation.

Appl. Math. Modelling, 1993, Vol. 17, December 619


_O.

(c)Soil Medium

Figure 2. Free body diagrams of the structural components of the liquid storage tank system.

2.1 Tank wall deflection function

The governing differential equation for the horizontal displacements, y(x), of the tank wall can be written as f0110ws:*

d4y(4 dX4 + 4P4y(x) = - y

f

where

D, = -J&t:

12(1 - v,z)

For most practical cases, the be considered large compared The solution of equation (1) expressed as*

(1)

(24 ,.(x)=~Cg2A.~/(BX)+c~~2(Px)--cliK(d-x)l (10)

(2’4

height of the wall can with the wall thickness. for such cases can be

2.2 Plate foundation deflection function

The plate foundation displacement function, w(r), is approximated by the following power series expansion:

w(r) = a i Ai r 0

2i

(11) i=O a

where n is the integer and the A,s are the unknown generalized coordinates. This displacement function must satisfy the folfowing boundary conditions applicable to the plate foundation-tank wall junction:

y(x) = - ~ ’ [IB~,_fiW + Qef2UWl V3D,

- 2 (d - x) f f

where

fi@x) = eCBX(cos /Ix - sin /?x) (4a)

f&?x) = e-OX cos /3x (Jb)

and M, and Q, denote the bending moment and shear force, respectively, at the junction of the tank wall and the plate foundation.

To simplify the computations, it is reasonable to assume that the stiffness of the plate foundation in the radial direction is large compared with its stiffness in the normal direction. This assumption implies that the radial stretching of the plate foundation is negligible compared with its normal deflections. It also implies that the horizontal displacement of the tank wall, at the junction with the plate foundation, can be neglected. This is expressed in a mathematical form by

Y(0) = 0 (5) Substituting equation (3) into equation (5) and solving for Q,, gives

Substituting back equation (6) into equation (3) leads to the following equation for y(x):

1 Y(X) = __

W3Dt BM,f3(Bx) + $ f2(8x) 1

where

- 2 (d - x) f t

(7)

f3(/Ix) = e-OX sin px (8)

To make the variables dimensionless, the following dimensionless quantities are defined:

xl, = g1 = 6(1 - vf)

%t; E t3 . ” = 2/?aD,’

4t; c1 =p; and c2 =pp 4b3a3D, E,t,ad

(9)

In view of equation (9), equation (7) can be written as follows:

CM,],=,= -Dp[$+:$]r=a= -Me (12a)

[Q,],=, = D, “B,:‘+ 5 $ - f $ 1 = 0 (12b) 1=11

WC)



component of the contact stress is present. Furthermore, the continuity condition assumed at the interface implies that the deformation of the soil surface can be rep- resented by the plate displacement function, w(r), given by equation (16). As shown by Sneddon,” for this situation the normal stresses at the interface can be uniquely determined from the theory of elasticity by considering the soil medium to be an isotropic elastic half space subjected, at its surface, to an axisymmetrical displacement field, w(r), for 0 < r 5 a. Using integral transforms, Sneddon has shown that the distribution of normal contact stresses, q(r), can be expressed as follows:

q(r) = - 1 LId a E

2 (1 - vf) r dr s

th(t) dt I J;“I”

where

D, = EPt:

12(1 - v;, (13)

Invoking the boundary conditions given by equations (12a)--(12c), the generalized coordinates A, _ 2, A, _ 1, and A, are expressed in terms of &!l, and the other generalized coordinates (i.e., Ais where i = 0, 1, 2, . . . , n - 3) by the following matrix equation:

F(1, n - 2) F(1, n - 1) F(1, n)

F(2, n - 2) F(2, n - 1) 172, n) F(3, n - 2) F(3, n - 1) F(3, n)

(14)

where N = n - 3, and the elements of the (3 x 3) coefficient matrix [F] and c3 are given by

F(1, i) = i(2i - 1 + vP) (15a)

F(2, i) = 2i (W

F(3, i) = i2 (i - 1) (15c)

c3 = c2 - ci (15d)

In equations (15a)-(15c) i may take any value from 1 to n. Solving equation (14) for An-2, A,_ 1, and A, and substituting the results into the plate foundation displacement function given by equation (11) yields

P2’ + i /Z(k, i)P2cnek) k=O 1

+ g&i, i p(k)p2’“-k’ + gc, i C(k)p2’“-k’ k=O k=O

where (16)

;i(k, i) = - i f(3 - k, 1)F(1, i) (174 I=1

p(k) = sl.03 - k> 1) + gzf(3 - k, 2) (1W

c(k) = f(3 - k, 2) (17c)

p2 a (1-W

and f( , ) denotes the elements of a (3 x 3) matrix

Cfl given by Cfl = VI -I, where [F] is the matrix on the left-hand side of equation (14). The variable p defined by equation (17d) is the dimensionless radial length.

2.3 Plate foundation-half space contact stress distribution

As mentioned earlier, because the plate foundation- soil interface is considered to be smooth, only the vertical

for 0 < r I a (18)

where

h(t) = E f s d,::“,

dr for 0 I t I a (19)

Noting that r = Pa (refer to equation (17d)), substituting equations (16) and (19) into equation (18) and evaluating the required integrals yields the following expression for the contact stress distribution, q(P), in terms of the independent generalized coordinates, ii?, and Ai, where i = 1,2, . . . , N:

4(P) = ES

’ { 2 Ai[io c.op’“] x(1 - vf) _Jm i=O

+ ClRe i: 5uP2u + CT3 i: r”P2u (20) u=o u=o

where t,(i), i,,, and ye. are dimensionless quantities and are defined by the following expressions:

5,tij = + ij 1% - 4 - 31!! CW!l’ >

[2(i - u)]!! [(2i - 1)!!12

+ i s(u, n - k)i(k, i) [2(n - k - u) - 3]!!

k=O [2(n - k - u)] ! !

[[2(n -k)]!!]’

x [[2(n - k) - l]!!]*

for i= 1,2,..., N and u=O,1,2 ,..., i (21a)

5,(i) = i s(u, n - k)A(k, i) [2(n - k - u) - 3]!!

k=O [2(n - k - u)]!!

[[2(n - k)]!!]”

’ [[2(n - k) - 1]!!12

for i = 1, 2,. . . , N and (i + 1) I u < (N + 1) (21b)

r,_,(i) = i s(l, k)i(k, i) c2(1 - k, - 31 !! k=O [2( 1 - k)] ! !

[[2(n - k)]!!12

’ [[2(n - k) - 1]!!12

for i = 1,2,. . . , N and I = 0, 1 (21~)



t,(O) = 0 for 24 = 1,2, . . . , n (214

50(O) = 1 (214

io = _ i cL(k) Mn-WI!! 1

k=O [2(n-k)-l]!! [2(n-k)-l]!! (210

i, = i s(u, n - kMk) [2(n - k - u) - 3]!!

k=O [2(n - k - u)]!!

[[2(n - k)]!!]”

’ [[2(n - k) - l]!!]’

for u= 1,2,...,(N+ 1) (21g)

[n_l = i s(l, Q(k) 12(’ - k, - 31!! k=O [2(1 - k)]!!

[[2(n - k)]!!]’ x [[2(n _ k) _ 1]M]2 for ’ = ” ’ W)

v], = [, when c(k) is used instead of p(k)

for t4 = 0, 1,2,. . . , n (21i)

s(u,j)= 1 ifu=j (2lj)

s(u,j)= -1 ifufj (21k)

j!!=lx3x5X ... x j for oddj and j 2 0 (211)

j!!=2x4x6x ... x j for even j and j 2 0 (21m)

j!! = 1 forjI0 (2ln)

2.4 Total potential energy functional

The total potential energy functional, U, of the tank wall-plate foundation-elastic half space system is evaluated as the sum of five contributions

u = u, + u, + U” + u, + Uh

where

(22)

U, = strain energy stored in the soil medium, U, = strain energy due to the plate foundation

bending, U, = work done by the vertical liquid pressure, U, = strain energy due to the tank wall flexure, and U,, = work done by the horizontal liquid pressure.

Computations for each of these terms are described in the following five subsections.

2.4.1 Strain energy stored in the soil medium

The strain energy stored in the soil medium is the work done by the contact stresses at the plate-soil interface. It is given by the following integral:

u, =f

2n (I

ss rq(r)w(r) dr d0 (23)

0 0

Introducing the dimensionless variable p defined by equation (17d), substituting w(p) and q(p) given by equations (16) and (20), respectively, into equation (23), and

evaluating the resulting integrals, yields

+sA, i ‘Ml(i) + gc, 5 441(i) i=O i=O

+g2A-tf,2C2, + g2A?,c3A, + g2c:lFl (24)

where xl(i,j), $1(i), $1(i), R,, A1, and r1 are all dimensionless quantities, defined in the Appendix by equations (Al)-(A6).

2.4.2 Strain energy due to the plate foundation bending

The strain energy stored in the plate foundation is given by

u, +

277 (I s sr r[V2w(r)12 0 0

- (25)

where

V2w(r) = 5 + i 2 (26)

In view of equations (16), (17d), and (20) U, can be expressed as

i 2 AiAjX2(i,j) i=CJ j=O

N N

+ SMe 2 Aih(i) + gC3 2 Ai42(i) i=O i=O

+ g21i?@2, + g2Mec3A2 + g2c:r2 1

(27)

where x2(i,j), ti2(i,j), 42(i, j), Q,, A2, and r2 are dimensionless quantities and are defined in the Appendix by equations (A7HA12).

2.4.3 Work done by the vertical liquid pressure

The intensity of the vertical liquid pressure acting on the plate foundation is constant and is given by

P = yd (28)

The work done by this pressure is given by 2n cl

U” = -

ss Pw(r)r dr de (29)

0 0

Substituting equations (16) and (17d) into equation (29) and performing the integrations over p and 8 gives

U, = --Pa3 i

i Aix3(i) + gli;i,A, + gc3r3 (30) i=O 1

where x3(i), A3, and r3 are dimensionless quantities defined in the Appendix by equations (Al3)-(A15).


2.4.4 Strain energy due to tank wallJEexure

The strain energy due to the flexural bending of the tank wall is given by

(31)

In view of equations (10) and (31), U, can be expressed as

u, = 7LD,g2[ti,2H1& + Hi,H2A, + r,] (32)

where the expressions for H, , H,, R,, A,, and I4 are dimensionless quantities, and are defined in the Appen- dix by equations (A16)-(A20).


in terms of the following additional dimensionless parameters:

“1 = tt. c(2 = d. K, = (1 E, !!! a’ a’ E, 0

3; a

K =(l -Vf’Er3. R = n f

K .

ES

f 1, p 6(1 - vi) ”

2.4.5 Work done by the horizontal liquid pressure

The work done by the horizontal liquid pressure acting against the tank wall is given by

2n d

U, = a ss

~(d - x)y(x) dx d0 (33) 0 0

In view of equations (3) and (33), U, can be expressed as

U, = rtPa3g[I\;;I,H3A, + I-,] (34)

where the expressions for the dimensionless quantities H,, A,, and IS are given in the Appendix by equations (A21)-(A23).

2.5 Determination of the unknown independent generalized coordinates

According to the principle of minimum potential energy, the independent generalized coordinates, Ai (where i=O,1,2 ,..., N) and ri;r,, should be so chosen that the value of the total potential energy functional is a relative minimum. Mathematically this is expressed by

au - = 0 for i = 0, 1,2, . . , N dA,

au

ari;l, -0 Wb)

Writing equation (35a) for each “i” and equation (35b) will result in a set of linear simultaneous equations that can be grouped in matrix form as

P’IV) = 14 (36)

where [V] is the known coefficient matrix of order (M x M), in which M = (N + l), {A} contains the unknown independent generalized coordinates and is a vector of order (M x l), and {B} is a known vector of order (M x 1). Noting that a(Ai)/aA, = 6, and LJ(A,A,)/aA, = AisjU + A,6,,, where aij is a Kronecker delta, the analytical expressions for the elements of the matrix [V] and the vector {B) can be derived. However, it is convenient to represent the results

(37)

Then the elements of the coefficient matrix [VI are given by

Vi, j) = IQ, i) = [x1(&j) + xl(j, 91 + : Cx2(kj)

+ x2(j, i)] for i, j = 0, 1, 2, . . . , N

but i, j # M (38a)

V(i, M) = V(M, i) = $I(i) + : G2(i)

for i=O,1,2 ,..., N

V(M,M)=2 [

Q, +$Cl2+$H,Q3 1 WW The elements of the known vector {B} are given by

B(i) = rcP(1 - VI)

g& S(i) for i = 1, 2, . . , N (39a)

B(M) = 7ccp(l - v,“)

g Es S(M) (39’4

where

S(i) = x3(i) - Ti 1 (404 S(M)= -[t(A1+$!A2)+A3+~2A~+~3A~]

(40b)

and the elements of the unknown vector {A} are related to the unknown generalized coordinates, A,, A,, A,, . . , A,, and the nondimensional edge moment by M, by

Ai = gA(i) for i = 0, 1, . . . , N @la)

Ii%, = A(M) (41b)

After the elements of matrix [I’] and vector {B) have been formulated for a problem, the unknown independent coordinates can be computed by inverting equation (36). Denoting

[u] = [VI-’ (42)



equations (41a) and (41b), respectively, can be written as

Ai = g E u(i, j)B(j) for i = 0, 1,2, . . , N (43a) j=O

fl, = 9 f 4M,jP(j) (43'3 j=O

Once the values of the Ais and Ml, are obtained, the tank wall displacement, y(X), the plate deflection, w(p), and the contact stress distribution, q(p), can be computed from equations (lo), (16), and (20), respectively, where X = x/d and p = r/a are the dimensionless length variables. For convenience in presenting the numerical results, these will be expressed in nondimensional form, y(X), W(p), and 4(p), respectively, as follows:

Y(X) = E,

Pa(1 - vz) Y(X) (44a)

*(PI = ES

Pa(1 - vf) W(P) (44’4

4(P) = f 4(P) (44c)

2.6 Evaluation of tank wall stress resultants

In the cylindrical tank wall, the longitudinal moment, M,, the circumferential moment, M,, the shear force, Q,, and the membrane force, N,, respectively, are computed from the following expressions:8

M&F; dM,. M, = v,M,; Q, = - dx , and

N, = _ Ett, y(x) a

(45)

For convenience in presenting the numerical results, the stress resultants given by equation (45) are expressed in nondimensional form as

H,(X) = & M,(X); Q,(X) = &Q,(x); and

~&X)=~N,W~ (46)

Substituting equation (10) into equation (45), and noting that x = Xd, the expressions for the dimensionless stress resultants can be expressed as follows:

a,(X) = $J2(BOX) 5 v(M,j)S(j) j=O

a1

- 2J30 f3(PoX)

Q,(X) = 7c[3(1 - $)]l/4 3 & CMBOX) + f3(/joX)l

jjjo vhj)W - A 2[3(1 - v:)]~‘~

fi@ox) W-4

m&O = -2q/1-v:)~ f3(BoW F vhj)W j=O

where

-_MPoW -x + 1 (47c)

Bo = C3(1 - W4~,lJcc, (48)

2.7 Evaluation of the plate foundationflexural moments

In the circular plate foundation, the radial moment, M,(r), and the tangential moment, M,(r), are computed from the following expressions,8 respectively:

M,(r)= -D,

MB(r) = -D, (494

(49’4

For convenience in presenting the numerical results, the stress resultants given by equations (49a) and (49b) are expressed in nondimensional form as

a,(p) = &M,(P) (50a)

@e(p) = + M,(p) (50b)

Substituting equation (16) into equations (49a) and (49b), and noting that r = pa, the dimensionless flexural moments can be expressed as

or = - zpf;lyv2) ,$ i(2i - 1 + vJA~~~‘-~ (5la) s 1 1

ho = - np:!v2, ,$ i[l + (2i - l)V,]Aip2’-2 s I 1

WV

3. Numerical results

To verify the analytical formulation presented in this paper, first a numerical example is solved and the results are compared with the finite element results presented by Booker and Small. l3 Then a parametric study is undertaken to investigate the effects of the basic geometric and material-related parameters that characterize the behavior of the cylindrical tank system. The effect of the variation of these parameters on the predicted dimensionless response defined in Section 2.6 is discussed next.

3.1 Numerical verification

The following numerical example presented by Booker and Small’ 3 was solved:

1. Liquid unit weight: y = 9.81 kN/m3



2. Tank wall properties: height d = 7.5 m thickness t, = 0.36 m Young’s modulus E, = 14 x lo4 MPa Poisson’s ratio U, = 0.0

3. Plate foundation properties: radius ap = 9.0 m thickness t, = 0.36 m Young’s modulus E, = 14 x lo4 MPa Poisson’s ratio vp = 0.0

4. Soil medium properties: Young’s modulus E, = 20 MPa Poisson’s ratio v, = 0.40

Although the values v, = 0.0 and vp = 0.0 are not real- istic, it will be shown later that these two parameters have little effect on the results obtained.

Solutions for this example were obtained by using the analytical method presented for a wide range of values of n, where n indicates the number of terms considered in the series expansion for the plate foundation displacement function, w(r), defined by equation (11). A “convergent solution” by the method presented in this paper is defined as the solution that uses the least number of terms and in which any addition of a term changes the results predicted by I 1%. Such a convergent solution was obtained for the plate foundation displacement function for n = 20 and for the contact stress distribution for n = 30. The results obtained for selected values of n and the finite element (FE) results reported by Booker and Smalli for the contact stress distribution, the plate foundation displacements, the plate foundation radial moment, the tank wall longitudinal bending moment, and the tank wall membrane forces are shown in Figures 3-7, respectively.

Figure 3 shows that near the plate center (p I 0.1) the convergent solution obtained by the method presented and the finite element solutions are in close agreement, with a difference of less than 2%.

The differential plate foundation displacement, w&), at any distance p from the plate center is defined as the difference [w(p) - w(l)], where w(1) is the displacement at the plate edge (where p = 1). A comparison of the

- “230

--- “Z20

80 . . . . FlNlTE ELEMENT (r&amnce 13)

. . . /

‘\ \ I. u \ l ,I \ \ /

Figure 3. Variation of contact stress distribution versus radical distance for the numerical example.

“‘20

--- n=, 5

. . . . FlNlTE ELEMENT (mfem”ce 13)

wd (mm)

0 > I”“““‘/““““‘I”“““‘1”“’ ‘f

0.0 0.2 0.4 0.6 0.8 7 .o P

Figure 4. Variation of plate foundation differential deflection versus radial distance for the numerical example.

- “>30

“CL?0

. . . . F,N,TE ELEMENThfemIC.

50 13.

. . . . APPROXMATION LINE

. .

-loo{ ,,,,,, ,,,,,,,,,, ,,,,,,, ,,,,, ,, ,, ,,,,, ,,,, ,,, ,,,,, 0.0 0.2 0.4 0.6 0.8 10

P

Figure 5. Variation of plate foundation radial moment vs. radial distance for the numerical example.

results for w,,(p) with the FE results is shown in Figure 4. The value of the differential plate foundation displacement at the center, w&O), is of particular interest for design. Figure 4 shows that near the plate center (for p I 0.1) the convergent solution obtained by the method presented correlates well with the FE solution, with a difference of less than 1%.

The variation of the plate foundation radial moment shown in Figure 5 shows that for lower values of n (n = 20) the results show some oscillating trend along the whole length of the radial distance p. It should be noted that the moment function is obtained by taking the second derivative of the plate deflection function. Be- cause the plate deflection function is approximate, the error in moments is much larger than for deflections. As n is increased beyond 20, the results become smoother and a convergent solution up to p I 0.75 is obtained for n = 30. The results varied slightly between 0.75 % p I 0.95. However, for-p > 0.95 and n = 40, the results still had some oscillating trend, and so this curve has been cut off at p = 0.95. Theoretically, if .n is increased indefinitely, this point should reach the plate edge (where p = l), but the computational round-off error may affect



- PRESENT WORK

-- CLASSlCAL THEORY hf-0 8)

7: . . . . FINITE ELEMENT (rat-e 13)

0- --.

,.““,“‘I”“““‘f

-25 0 25 50 75 100

,A=( kN- mm)

Figure 6. Variation of tank wall longitudinal moment vs. vertical distance for the numerical example.

-PRESENT WORK

---CLASSICAL THEORY (referrme 8)

. . . UNITE ELEMENT ~rehnnce 13)

N (kN/n, 1

Figure 7. Variation of tank wall membrane force vs. vertical distance for the numerical example.

the results. This difficulty can be overcome by noticing that the plots obtained for n 2 30 exhibit a specific trend near the plate edge (p 2 0.75). Indeed, these curves, which coincide in the region p = 0 to about p = 0.75, split into distinct branches starting at about p = 0.75 and gradu- ally move up as n increases from 30 to 40. Following this trend, the final plot that would be obtained for larger n can be reasonably approximated by the dotted line extended up to the plate edge, as shown in Figure 5. This extension is made on the smooth part of the curve just before the splitting point. This way, a rational approximation to evaluate the edge moment can be obtained.

Because high concentrations of moment often occur at the plate foundation center and at the plate foundation-tank wall junction, the prediction of the central moment, M,(O), and the edge moment, M,, is of particular interest in structural analysis and design. Figure 5 shows that both the central and the edge moments predicted by the present analysis are somewhat lower than the FE values. The edge moment (M, = 85 kN - m/m) obtained from the present analysis (using the approximation line

shown in Figure 5) is 34% more than the value (M, = 56 kN - m/m) predicted by the classical treatment’-” of the storage tank. Thus, the classical theory, which assumes the tank wall to be fixed at the base, gives unconservative results. This justifies the necessity to include the interaction between the plate foundation and the tank wall in the analysis.

Knowing the edge moment, M,, the internal forces of the tank wall can be evaluated using the classical theory of shell analysis with appropriate boundary conditions. The results for the longitudinal bending moment, M,, and the membrane force, N,, are compared in Figures 6 and 7, respectively. It is observed from Figure 6 that the maximum longitudinal moment in the tank wall occurs at the junction with the plate foundation (at x = 0). The membrane forces (see Figure 7) show good agreement with the results obtained by FE and classical theory. The maximum value (at x = 3m) predicted by the FE analysis is reproduced within 2% by the method presented here and 6% by the classical theory. The discrepancy in the behavior of N, near x = 0, observed in Figure 7, could be attributed to the simplification made in the present work and in the classical treatment (but not in the FE analysis), by assuming that the horizontal displacement of the tank wall at x = 0 is equal to zero. This assumption affects the membrane force near the plate-wall junction.

3.2 Parametric study

As can be seen in Section 2.6 (equation (37)) the primary dimensionless variables, 4(p), W(p), and M,(p), are functions of the following parameters: CQ , CQ, ?;, and KC. Also note that the dimensionless edge moment, M, = -M,(O). Because the maximum values of the afore- mentioned dimensionless variables are of interest to the designer, in this section the effect of the following variables on the maximum values predicted is presented: c1 1, a2, 7;, K,, v,, vsr and E,. In all these analyses n = 30 was used.

To determine the range of variation of the soil-related parameters, the soil medium is considered to vary from soft clay to rock. This domain can be covered by varying E, between 1 ksi and 1500 ksi and varying v, between 0.25 and 0.40. The most practical estimate for E, is to consider concrete or steel for the tank wall material, which gives E, = 3000 ksi or E, = 29,000 ksi, respectively. In view of these values, the approximate limits of variation of z are obtained as 1 and 30,000. The results obtained for these two extreme values of 7; are plotted against the plate relative rigidity, K,, and presented in Figures 8(a)-8(c). These figures show that changing ?; from 1 to 30,000 results in a variation of less than 2% for the contact stress and the plate deflection, but about 10% for the plate radial moment. Thus, to investigate the effect of other parameters, the parameter IT; was kept constant and equal to an intermediate value: IT; = 1000.

The parameter CI~ represents the tank wall thickness normalized by the plate foundation radius. The computations are performed for c1i = 0, 0.01, and 0.1, and the results are presented in Figures 9(at9(c). The value



1.2

1.1 --- T

% j--,

Tt = 30,000

t=l

0.4-a I,'-

-3 -2 -1 0 1 2 3

'o910 KP

(a)ollcontactstressatplatetandatbncenbBl

Tt = 30,000

--- Tt = 1

-3 -2 -1 0 1 2 3

'O9lO KP

(b) On plats fomdaUoa caliladailwtion

- 0.08' Tt = 30,000

---T ~1 t

0.06-

%

Cc) on fatate fomdamnc~mmmnt

Figure 8. Effect of Tt and plate relative rigidity, Kp, a2 = 1 .O, and Y,, = Ye = 0.30.

for a, = 0.1,

u1 = 0 means that the tank wall does not exist; therefore, this corresponds to the case of an isolated circular plate resting on the soil medium. Figures 9(a) and 9(b) show that the contact stress and the deflection at the plate center are affected insignificantly (less than 1%) by CI~ for a very rigid plate (log K, 2 2) or a very flexible plate (log K, I -2). Therefore, for these two limiting cases, the whole storage tank system behaves the same way as if the tank wall did not exist. Also, as it can be seen from

- a1 = 0.10

1.2 1 --- a , = 0.01

1.1 ______- "1 = 0.0 0

1.0

0.9

% 0.8

0.7

0.6

0.5

0.41, , , , ,

-3 -2 -1 0 1 2 3

109 10 KP

- a. = 0.10 2.2

---a, = 0.01 2.1

_______ Q =

$T,-__;

7

1.5-

1.4A_,

-0 -2 -1 0 1 2 3

'O910 KP

-01 zo.10

--- 01 = 0.01

-____-- =1 = 0.00

0.08 -1 Xc

~____ ____------ ,’

0.04

_#:.i,

I' _---

/J1_

-3 -2 -1 0 1 2 3

‘09 Kp 10

(c)on lhte lundath~mmteni

Figure 9. Effect of cq and plate relative rigidity, Kp, for a2 = 1 .OO, Tt = 1000, and vP = vy = 0.30.



these figures, the contact stress and the plate deflection at the plate center converge toward the respective values obtained by the classical theory of elasticity. For the intermediate values of KP( -2 I log K, I 2), the parameter a, can significantly affect the results. The maximum difference between the results obtained for ~1~ = 0 and ~1~ = 0.1 occurs at log K, = -0.5 and is in the order of 25%.

For the plate central moment (Figure 9(c)), M,, the effect of ~1~ is insignificant (less than 1%) for very flexible plate (log K, 5 -2). But changing c(~ from 0 to 0.1 rapidly increases the difference between the two values of M, from 3% to 60%, as the plate relative rigidity increases in the intermediate range from log K, = - 1 to log K, = 1, respectively. For a rigid plate (log K, 2 1) the effect of a, stabilizes; the results obtained for a, = 0 are more than double those obtained for a, = 0.1. There- fore, if the tank wall is completely ignored in the analysis, by taking CI~ = 0, the plate central moment may be greatly overestimated in such cases.

The parameter c(~ represents the height of the liquid in the tank normalized by the plate foundation radius. For most practical cases, the value of CQ can vary from 0.5 to 2. The computations are performed for these two extreme values and for an intermediate relative rigidity of the plate foundation (log K, = -0.5). The results are plotted in Figures IO(a)-IO(c). From these figures it can be seen that the effect of the parameter c(~ on the primary variables q, w, and M, is relatively insignificant (< 1%).

In general, the materials used for the tank wall and the plate foundation are concrete or steel, for which Poisson’s ratio can be taken between 0.15 and 0.30. Here again, the computations are performed for the plate relative rigidity (log K, = -0.5). The results obtained for vp = 0.30 and two values of v,, 0.15 and 0.30, are plotted in Figures 12(a)-21(c). It is observed that the effect of v, on the primary variables q, w, and M, is negligible (less than 0.1%). The results obtained for v, = 0.30 and two values of vp, 0.15 and 0.30, are plotted in Figures 12(a)-12(c). These figures show that the effect of vp on the primary variables q, w, and M, is somewhat more than that of v, but this effect is still insignificant (<l%).

In the analytical formulation presented in this paper, the elastic properties of soil are defined by Young’s modulus, E,, and Poisson’s ratio, v,. These two quantities do not appear in one unique parameter, they are expli- citly present in two basic parameters, T and K,, and in the plate deflection function, w(p). In order to investigate the effect of the soil elastic properties on the plate foundation behavior, a distinct parameter is defined as follows:

(52)

As previously mentioned, if E, is varied from 1 ksi to 1500 ksi and v, is varied from 0.25 to 0.40, the parameter 4, will vary between 10m4 and 10, respectively, where the lower limit is applicable for very hard rock and the upper limit is for very soft clay. The two basic para-

0.6 -

0.5

0.4 I I I I I

0.0 0.2 0.4 0.6 0.8 1.0

P

(a) On ccntact atreaa diaWutkn

2.2

2.1

2.0 1

a, I 2.0 0

--- Q2 = 0.5 0

0.0 0.2 0.4 0.6 0.8 1.0

P

(b) On plete tcmhtbn ddtectIon futctlcn

0.1 0 3 - 4,~2.00

--- a2 = 0.50

0.0 5

I:...1 - 0.0 5

I I I I I 0.0 0.2 0.4 0.6 0.8 1.0

P

(c)On plate focdatimradiafmomenf

Figure 10. Effect of at and radial distance, p, for CC, = 0.10, Tt = 1000, log Kp = -0.50, and v,, = vy = 0.30.

meters, T, and K,, can be related, respectively, to c#J,, as follows :

(534

(5W

Therefore, when 4, varies both parameters T, and K, vary simultaneously.

The computations are performed for the following data: CI~ = 0.01, a2 = 1.0, E, = 3000 ksi, E, = 29,000 ksi, vp = vt = 0.30, t, = 15 in., and a = 10 ft. The results



(a) On contact stress dlstrlbutlon

F 1.9

1 .6

ii r

1 .4A4

0.0 0.2 0.4 0.6 0.8 1 .o

P

(b) On plate toundatlon deflection function

0.10

I

- yt = 0.30

= 0.15 --- y t

0.05

();~-\ \

-0.05 I I I I I

0.0 0.2 0.4 0.6 0.6 1.0

P

(c) On plate toundatkn radial moment (cl On plate fomdatlon radial moment

Figure 11. Effect of vf and radial distance, p, for cq = 0.10, a2 = Figure 12. Effect of vp and radial distance, p, for a, = 0.10. 3~~ = 1 .OO, T1 = 1000, log Kp = -0.50, and vp = Y, = 0.30. 1 .OO, Tr = 1000, log Kp = -0.50, and vf = 0.30.

obtained for this example are plotted in Figures 13(ak M, attains a maximum ;alue for soft clay. The explana- 23(c). Figure 13(a) shows that the contact stress at the tion of these results is that on rocky soil the plate plate center, q0 = q(O), is equal to the intensity of the foundation does not experience any noticeable bending; applied load, yd, for hard rock, and equal to yd/2 for soft it remains almost horizontal. This implies uniform soil clay. Figures 13(b) and 13(c) show that the central plate reaction underneath the plate foundation as well as very deflection, w. = w(O), and the central plate moment, small bending moment in the plate foundation. Con- MO = M,(O), are almost zero for rock and rapidly in- versely, on clayey soil, the plate foundation will undergo crease as the soil medium changes to clay. While the a significant bending, which will cause large moments to deflection, w,,, keeps on increasing rapidly, the moment develop at the plate center.

0.63

(a) On contact stress dbtrlbutbn

F

jr- y--y-y o.la l1o

0.0 0.2 0.4 0.6

P

(b) On plate bundatbn deflectIon fmctkm

0.1 0 1 - YP 1 0.30

i

---

0.0 5

“p = 0.15

, %

-0.05 o.oo!,!, 0.0 0.2 0.4 0.6 0.6 1.0

P

Appt. Math. Modelling. 1993, Vol. 17, December 629

Analysis of fluid storage ranks: A. R. Kukreri et al.

5

Rock

qo (ksi) xl0 3

L Clay

2L -4 -3 -2 -1 0 1

‘o!3,0 0s

(dG,OOllt&~~t~fiMdatbn

150

100 3

w. (in) x103 Clay

50-

IJ , r Rock

0

-4 -3 -2 -1 0 1

‘-al0 9s

(bfonplatotomdamncambal~

m,(k-in/in)

3-J

2 Clay

l-

0 I I I I 1

-4 -3 -2 -1 0 1

(c) on p(atcr tandatlon~rmmam

Figure 13. Effect of soil elastic properties parameter 4,

4. Conclusion

An analytical procedure is presented to predict the behavior of a liquid storage tank resting on an elastic soil medium, which is modelled as an isotropic elastic half

space. The procedure accounts for the interaction between the tank wall and the plate foundation, as well as the interaction between the plate foundation and the soil medium. Analytical expressions are obtained for all displacements and internal stress resultants of interest.

Results of a numerical example showed that the convergent solution obtained by the analytical formulation presented here compares reasonably well with the finite element solution. The analytical formulation presented in this paper has advantages over other numerical techniques, such as the finite element method in that it can be used for any geometric configuration and material properties with very little computational effort. The formulation can be conveniently automated on a PC.

A parametric study has also been conducted to investigate the effects of the basic (geometric and material) parameters, and of the soil elastic properties, on the soil-foundation-tank behavior. The graphical results presented here can be used to estimate the flexural response of cylindrical tank foundations of practical importance, including interaction effects.

References

1.

2.

3.

4.

5.

6.

I.

8.

9.

10.

11.

12.

13.

14.

15.

16.

17.

Borowicka, H. Influence of rigidity of a circular foundation slab on the distribution of pressure over the contact surface. Proc. of the 1st Int. Conf: on Soil Mech. and Foundation Eng. 1936, 2, 144-149 Zemochkin, B. N. Analysis of circular plates on elastic foundation. Mask. Izd. Inzh. Akad. Moscow, 1939 Ishkova, A. G. Exact solution of the problem of a circular plate in bending on the elastic halfspace under the action of a uniformly distributed anti-symmetrical load. Dokl. Akad. Nauk. 1947, 4, 129-132 Brown, P. T. Numerical analyses of uniformly loaded circular rafts on deep elastic foundations. Geotechnique 1969, 19, 301-306 Selvadurai, A. P. S. Elastic analysis of soil-foundation interaction. Developments in Geotechnical Engineering. Vol. 11. Elsevier Scien- tific Pub]. Co., New York, 1979, pp. 281-351 and 375406 Faruque, M. 0. and Zaman, M. M. Approximate analysis of uniformly loaded circular plates on isotropic elastic half-space. Proc. of the 9th Congress of the Nat ‘1. Acad. of Eng. of Mexico. Leon, Mexico, 1983, pp. 271-276 Zaman, M. M., Kukreti, A. R. and Issa, A. Analysis of circular plate-elastic half-space interaction using an energy approach. Appl. Math. Modelling 1988, 12, 285-292 Timoshenko, S. P. and Woinowsky-Krieger, S. Theory of Plates and Shells, 2nd ed. McGraw-Hill Book Co., New York, 1959 Krauss, H. Thin Elastic Shells. John Wiley and Sons, New York. 1959 Baker, E. H., Kovalevsky, L. and Risk, F. L. Structural Analysis of Shells. McGraw-Hill Book Co., New York, 1972 Creasy, L. R. Prestressed Concrete Cylindrical Tanks. John Wile) and Sons, New York, 1961 Manning, G. P. Reservoirs and Tanks. Concrete Publication, Ltd. London, 1967 Booker, J. R. and Small, J. C. The analysis of liquid storage tank! on deep elastic foundations. Int. J. Numer. Anal. Methods Gee. mech. 1983, 7, 187-207 Cheung, Y. K. and Zienkiewicz, 0. C. Plates and tanks on elastic foundations-an application of finite element method. Int. J Solids Struct. 1965, 1, 451-461 Smith, I. M. A finite element approach to elastic soil-structure interaction. Canadian Geotech. J. 1970, 7, 95-105 Mahmood, I. U. Finite element analysis of cylindrical tan1 foundation resting on isotropic soil medium including soil- structure interaction. M.S. Thesis, University of Oklahoma, 1984 Sneddon, I. N. Fourier Transforms. McGraw-Hill Book Co., Nev York. 1951


Appendix

The dimensionless quantities xl(i,j), t+bl(i), dl(i), Q,, AI, and rI appearing in equation (24) are given by, respectively,

xl(i,j) = i t,(i) 112(’ +‘)I!! + i n(k,j)

Id=0 [2(u + j) + l]!! k=O

[2(u + II - k)]!!

’ [2(u + n - k) + l]!! for i,j=O,1,2 ,..., N

(Al) Ic/I(i) = i (, C2(u + ill!!

u=o [2(u + i) + l] ! ! + j. C4U(i)dk)

+ i, d(k, 91 [2(u + n - k)]!!

[2(u + n - k) + l]!!

for i = 0, 1,2,. . , N (A2)

4I(i) = i vlu 2 C2(u + W u=o [2(u + i) + l]! ! + ,go csu(ik(k)

[2(u + n - k)]!! ___.__ +Ylu/Z(k’ i)l [2(u + y1 - k) + l]!!

for i = 0, 1, 2, . . .

a, = i i i,pL(kI Mu+n-VI!! N 643)

644) u=O k=O [2(u+n-k)+ l]!!

A1 = f i [&c(k) + q,(k)] x c2(u + ’ - k)l!! u=O k=O [2(u + n - k) + l]!!

(W

rI= i 2 ,,/(k) C2(u+n-k)]!!

[2(u + n - k) + l]!! (‘46) u=O k=O

The dimensionless quantities x2(i, j), $2(i, j), 42(i), R,, A, and r2 appearing in equation (27) are given by, respectively,

x2&j) = 8 i

2i2j2 - (1 - vJij(2j - 1)

2(i + j - 1)

+ i I(k, i)

k=O [

2j2(n - k)2

(i+n-k-1)

- (1 - v,)j(n - k) 1 +ii 4% W, j) k=O I=o 2(2n - k - I- 1)

x [2(n - Q2(n - I)’ - (1 - v&n - k)

x (n - 1)(2n - 21 - l)] i

for i, j = 0, 1,2,. . . , N (A7)

t,k2(i) = 8 i [

$ p(k) 2i2(n - k)2 - (1 - v&n - k) k=O (i + n - 1) 1 + i i Atk. iJdlj 2(n - k)‘(n - O2 k=O I=0 (2n - k - I- 1)


-(l - v,)(n - k)(n - I) 11 for i = 0, 1,2,. . . , N (A8)

42(i) = 8 i c(k) i [

2i2(n - k)2 k=O (i+n-k-l)

- (1 - v,)i(n - k) 1 + i i Ack, i+clI 2b - W2b - II2 k=O i=O [ (2n - k - 1 - 1)

- (1 - v&n - k)(n - 1) 31 for i = 0, 1,2,. . . , N (A9)

02 = 4k$o ,io (2n !‘,““‘:‘_ 1) C2(n - k)2(n - 0’

x (1 - V&I - k)(n - 1)(2n - 2k - l)] (AlO)

A2 = 8 f: ; p(k)c(l) i k=O I=0 [

;;; 1 ;?lI;;

- (1 - v&n - k)(n - 1) II (All) r2 = 4 k$‘o [to (2n T’,““:’ _ 1) C2(n - k)2(n - U2

- (1 - V&I - k)(n - 1)(2n - 2k - l)] (A12)

The dimensionless quantities x3(i), A3 and r3 appearing in equation (30) are given by, respectively,

x3(i) = i:i + ,io (n ““: il 1)

for i = 0, 1,2 ,..., N (A13)

A3= i p(k)

k=O (n - k + 1)

r3= f: c(k) k=O (n - k + 1)

6414)

(A19

The quantities H,, H,, Sz,, A4, and r4 appearing in equation (32) are given by, respectively,

H, = ;/?ug; (‘416)

H2 = ,Cw2cl 6417)

l-& = -[f,(2/hq + 2C20d - 33 6418)

& = f2W4 + f&Y4 - 1 6419) r4 = &c:[f,(2pd) - 2e-2Bd + 11 6420)

The quantities H,, AS, and TS appearing in equation (34) are given by, respectively,

H3 =g$ 1

As = 1 - E Cl - Mwl

6421)

6422)

rs =c1 1 1

(Pa)’ 1 - 2 mw - z uw2 1

(A23)


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